Iwasawa theory and localized Bost-Connes systems: there are no p-adic KMS states, and where Leopoldt actually enters

Four obstructions kill any p-adic Bost-Connes thermodynamics: C_p admits no order, so there is no positivity and no state space; the dynamics is undefined at p because N(p) is not a p-adic unit; p-adic Dirichlet series have no abscissa, every term having absolute value one; and the proposed element cannot generate the characteristic ideal of Iwasawa theory, being a unit already for K = Q with chi(p) of order prime to p, and depending only on the decomposition data of p. Leopoldt enters the programme through the isotropy computation of Paper II (its Proposition 2.2 and Remark 2.3), not through a p-adic state space.